The Existence of Solutions to the Nonhomogeneous A -Harmonic Equations with Variable Exponent

Abstract

We first discuss the existence and uniqueness of weak solution for the obstacle problem of the nonhomogeneous $A$-harmonic equation with variable exponent, and then we obtain the existence of the solutions of the equation $d^{\mathrm{\star }}A(x,d\omega )=B(x,d\omega )$ in the weighted variable exponent Sobolev space $W_d^{p(x)}(\mathrm{\Omega },{\mathrm{\Lambda }}^l,\mu ).$